Neutral GeometryOctober 25, 2009c 2009 Charles DelmanTaking Stock: where we have been; whe re we are goingSet Theory&LogicTerms ofGeometry:points, lines,incidence,betweenness,congruence.IncidenceAxiomsPropositionsofIncidenceGeometryBetweennessAxiomsTheoremsTheoremsCongruenceAxiomsModelsPhilosophicalIssuesAffine &ProjectiveGeometryc 2009 Charles Delman 1In neutral geometry we assume the first threeincidence axiomsv Axiom I-1: For every point P and for every point Q that is distinct fromP , there is a unique line l incident with P and Q.v Axiom I-2: For every line l there exist (at least) two points incidentwith l.v Axiom I-3: There exist three distinct noncollinear points.v Remark. Any additional assumption m ade about the exis tence of parallellines, such as the elliptic, Euclidean, or hyperbolic parallel property wouldalso be an incidence axiom. That we are not making any such assumptionis the reason the theory we are currently studying is c alled “neutral.” Itis neutral on the question of parallelism.c 2009 Charles Delman 2Using only the three axioms above (in fact, using only I-1 and I-3), wehave so far proven the following propositions:v Proposition 2.1 If l and m are distinct lines that are not parallel, thenl and m have a unique point in common.v Proposition 2.2 There exist three distinct lines that are not concurrent.v Proposition 2.3 For every line, there is at le ast one point not incidentwith it.v Proposition 2.4 For every point, there is at least one line not incidentwith it.v Proposition 2.5 For every point, there are at least two lines incidentwith it.c 2009 Charles Delman 3Betweenness Axiomsv Recall that P ∗ Q ∗ R means that point Q is between points P and R.v Axiom B-1: If P ∗Q ∗R, then P , Q, and R are distinct, collinear points,and R ∗ Q ∗ P .v Axiom B-2: Given points Q and S, there are points P , R, and Tincident with line←→QS such that P ∗ Q ∗ S, Q ∗ R ∗ S, and Q ∗ S ∗ T .v Remark. Axiom B-2 does not assert P ∗ Q ∗ R or any other relationshipamong the points that does not involve both Q and S. The obviousadditional relationships are true but must (and will) be proven!v Axiom B-3: If P , Q, and R are distinct collinear points, then one andonly one of them is between the other two.c 2009 Charles Delman 4These three axioms suffice to prove:v Proposition 3.1 Let A and B be distinct points. Then (i)−−→AB ∩−−→BA =AB, and (ii)−−→AB ∪−−→BA = {←→AB} (where {←→AB} denotes the set of pointslying on line←→AB).v It will be useful to first prove the following lemmas:Lemma 1. Let A and B be distinct points. Segment AB is equal tosegment BA.Lemma 2. Let A and B be distinct points. The following three sets ofpoints on line←→AB are disjoint: {P ∈ P : P ∗ A ∗ B}, segment AB,{P ∈ P : A ∗ B ∗ P }.c 2009 Charles Delman 5Sides of a Linev To formulate the last axiom of betweenness, we need a definition.Definition: Given a line l and two distinct points P and Q that are notincident with l, P and Q are on the same side of line l if . . . (Exercise!).c 2009 Charles Delman 6v Definition: Given a line l and two points P and Q that are not incidentwith l, P and Q are on the same side of line l if P = Q or if no pointin segment P Q is incident with l. If P and Q are not on the same sideof l, they are said to be on opposite sides of l.v Axiom B-4 (Plane Separation) For every line l:(i) The relation of being on the same side of l is transitive.(ii) For any three points P , Q, and R not lying on l, if P and Q are onopposite sides of l and Q and R are on opposite sides of l, the P and Rare on the same side of l.Corollary (iii). If P and Q are on opposite sides of l and Q and R areon the same side of l, then P and R are on opposite sides of l.Lemma 3. The relation of being on the same side is an equivalencerelation.c 2009 Charles Delman 7What Axiom B-4 Means Intuitively (i)v Statement (i) ensures that our geometry is two-dimensional:PQRc 2009 Charles Delman 8What Axiom B-4 Means Intuitively (ii)v Statement (ii) ensures there are no “singular” lines at which more thantwo half-planes meet.PQRc 2009 Charles Delman 9Half-planesv Notation. Let [P ]ldenote the equivalence class of P . This notation,different from that used in the text, recognizes that each distinct linedetermines a different equivalence relation. The equivalence class [P ]lis often called the half-plane bounded by l that contains P . (The textuses HPto denote this half-plane, with the specific line l bounding thehalf-plane understood from c ontext.)v Proposition 3.2 Every line bounds exactly two distinct (and disjoint)half-planesThe fact that distinct half-planes are disjoint follows from Lemma 3;distinct equivalence classes are always disjoint. It remains to prove thatthere are exactly two.c 2009 Charles Delman 10Review of Betweenness Axioms, Definitions,& Propositionsv Axiom B-1: If P ∗Q ∗R, then P , Q, and R are distinct, collinear points,and R ∗ Q ∗ P .v Axiom B-2: Given points Q and S, there are points P , R, and Tincident with line←→QS such that P ∗ Q ∗ S, Q ∗ R ∗ S, and Q ∗ S ∗ T .v Axiom B-3: If P , Q, and R are distinct collinear points, then one andonly one of them is between the other two.v Proposition 3.1 Let A and B be distinct points. Then (i)−−→AB ∩−−→BA =AB, and (ii)−−→AB ∪−−→BA = {←→AB}.c 2009 Charles Delman 11v Definition: Given a line l and two points P and Q that are not incidentwith l, P and Q are on the same side of line l if P = Q or if no pointin segment P Q is incident with l. If P and Q are not on the same sideof l, they are said to be on opposite sides of l.v Axiom B-4 For every line l:(i) The relation of being on the same side of l is transitive.(ii) For any three points P , Q, and R not lying on l, if P and Q are onopposite sides of l and Q and R are on opposite sides of l, the P and Rare on the same side of l.Corollary (iii). If P and Q are on opposite sides of l and Q and R areon the same side of l, then P and R are on opposite sides of l.v Proposition 3.2 Every line bounds exactly two distinct (and disjoint)half-planes).c 2009 Charles Delman 12v Proposition 3.3 Given A ∗ B ∗ C and A ∗ C ∗ D, then B ∗ C ∗ D andA ∗ B ∗ D.Corollary. Given A ∗B ∗C and B ∗C ∗D, then A ∗B ∗D and A∗C ∗D.v Proposition 3.4 Given A ∗ B ∗ C, let P be a point on the line (given byAxiom B-1) through A, B, and